3.8 \(\int \frac{1}{a+b e^{-c-d x}} \, dx\)

Optimal. Leaf size=28 \[ \frac{\log \left (a+b e^{-c-d x}\right )}{a d}+\frac{x}{a} \]

[Out]

x/a + Log[a + b*E^(-c - d*x)]/(a*d)

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Rubi [A]  time = 0.022578, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2282, 36, 29, 31} \[ \frac{\log \left (a+b e^{-c-d x}\right )}{a d}+\frac{x}{a} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*E^(-c - d*x))^(-1),x]

[Out]

x/a + Log[a + b*E^(-c - d*x)]/(a*d)

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{1}{a+b e^{-c-d x}} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1}{x (a+b x)} \, dx,x,e^{-c-d x}\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,e^{-c-d x}\right )}{a d}+\frac{b \operatorname{Subst}\left (\int \frac{1}{a+b x} \, dx,x,e^{-c-d x}\right )}{a d}\\ &=\frac{x}{a}+\frac{\log \left (a+b e^{-c-d x}\right )}{a d}\\ \end{align*}

Mathematica [A]  time = 0.0147213, size = 19, normalized size = 0.68 \[ \frac{\log \left (a e^{c+d x}+b\right )}{a d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*E^(-c - d*x))^(-1),x]

[Out]

Log[b + a*E^(c + d*x)]/(a*d)

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Maple [A]  time = 0.003, size = 41, normalized size = 1.5 \begin{align*} -{\frac{\ln \left ({{\rm e}^{-dx-c}} \right ) }{ad}}+{\frac{\ln \left ( a+b{{\rm e}^{-dx-c}} \right ) }{ad}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*exp(-d*x-c)),x)

[Out]

-1/d/a*ln(exp(-d*x-c))+ln(a+b*exp(-d*x-c))/a/d

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Maxima [A]  time = 1.10787, size = 46, normalized size = 1.64 \begin{align*} \frac{d x + c}{a d} + \frac{\log \left (b e^{\left (-d x - c\right )} + a\right )}{a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*exp(-d*x-c)),x, algorithm="maxima")

[Out]

(d*x + c)/(a*d) + log(b*e^(-d*x - c) + a)/(a*d)

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Fricas [A]  time = 1.46414, size = 53, normalized size = 1.89 \begin{align*} \frac{d x + \log \left (b e^{\left (-d x - c\right )} + a\right )}{a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*exp(-d*x-c)),x, algorithm="fricas")

[Out]

(d*x + log(b*e^(-d*x - c) + a))/(a*d)

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Sympy [A]  time = 0.133725, size = 19, normalized size = 0.68 \begin{align*} \frac{x}{a} + \frac{\log{\left (\frac{a}{b} + e^{- c - d x} \right )}}{a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*exp(-d*x-c)),x)

[Out]

x/a + log(a/b + exp(-c - d*x))/(a*d)

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Giac [A]  time = 1.27964, size = 47, normalized size = 1.68 \begin{align*} \frac{d x + c}{a d} + \frac{\log \left ({\left | b e^{\left (-d x - c\right )} + a \right |}\right )}{a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*exp(-d*x-c)),x, algorithm="giac")

[Out]

(d*x + c)/(a*d) + log(abs(b*e^(-d*x - c) + a))/(a*d)